There are 2 known properties in DTFT:
- Time shift corresponds to multiplication with complex exponential
$$x[n-k] \rightarrow e^{-j\omega k}\cdot X_{2\pi}(\omega) $$ - Multiplication in time corresponds to circular convolution
$$x[n]\cdot y[n] \rightarrow{ {\frac {1}{2\pi }}\int _{-\pi }^{\pi }X_{2\pi }(\nu )\cdot Y_{2\pi }(\omega -\nu )d\nu \!} $$
Now, combining those two we get: $$x[n]\cdot x[n-k] \rightarrow{ {\frac {1}{2\pi }}\int _{-\pi }^{\pi }e^{-j\nu k}X_{2\pi }(\nu )\cdot X_{2\pi }(\omega -\nu )d\nu \!} $$
Is there a way to extract the exponential from the integral or at least to shape the integral somehow so that I can first separately calculate the circular convolution of the signal with itself and then do something with the exponential?